On log del Pezzo surfaces in large characteristic

Abstract

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally F-regular or it admits a log resolution which is liftable to characteristic zero. As a consequence, we prove the Kawamata-Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.Comment: v2: final version. To appear in Compositio Mathematic

New directions in the Minimal Model Program

“…On the other hand, klt singularities are not necessarily strongly F-regular (e.g. see [18,19]) and, therefore, we cannot consider the category of strongly F-regular pairs as a good candidate of the vertices for our directed graph. Indeed, it is easy to construct examples of strongly F-regular pairs such that their minimal model is not strongly F-regular.…”

On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

“…The situation simplifies in large characteristics. In [6] the authors prove the existence of an integer p 0 such that over an algebraically closed field of characteristic p > p 0 every log del Pezzo surface satisfies Kawamata-Viehweg vanishing theorem. Finding an effective bound for this p 0 is a central open question in positive characteristic birational geometry.…”

“…Finding an effective bound for this p 0 is a central open question in positive characteristic birational geometry. For example over an algebraically closed field of characteristic p > max{5, p 0 } a three The construction of the integer p 0 in [6] is implicit. During the course of the proof of [6, Theorem 1.1] the authors consider log del Pezzo surfaces belonging to a bounded family over Spec(Z).…”

“…From this classification it follows that all log del Pezzo surfaces of Picard rank one over an algebraically closed field of characteristic p > 5 admit a log resolution that lifts to W 2 (k) [17,Theorem 7.2]. It therefore follows from his work that Kodaira vanishing holds for log del Pezzo surfaces of Picard rank one over an algebraically closed field of characteristic p > 5 (see [6,Lemma 6.1]). Combining his result with the techniques exploited in this paper, one can prove the Kodaira vanishing theorem on a log del Pezzo surface over an algebraically closed field of characteristic p > 5.…”

Klt del Pezzo surfaces which are not Globally F-split

“…Note that, if X is a smooth del Pezzo surface over an algebraically closed field of characteristic p > 5, then X is globally F -regular [Har98a, Example 5.5]. In [CTW15], we show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic which is not globally F -regular, admits a log resolution which is liftable to characteristic zero. By considering the cone over a klt del Pezzo surface which is not globally F -split, as in Theorem 1.1, we obtain a three-dimensional klt singularity in arbitrary characteristic which is not F -pure.…”